Multiple-choice backpack task - dealing with complementary and alternative assets in strategic investment decisions

Classification

Today, companies are rarely faced with the decision of whether to invest, but rather how to optimally distribute limited funds among several interdependent options. Investments are neither isolated nor can they be combined at will. Rather, they exist:

• alternative assets, of which only one option may be chosen (either-or),
• complementary assets whose value only arises in combination (both-and).

It is precisely this reality that can be described precisely using a classic optimization model: the multiple-choice knapsack problem (MCKP).

1. From the simple budget decision to the structural question

The classic knapsack problem answers the question: Which objects do I pack in a backpack with limited weight to achieve maximum value?

In business practice, however, this model is not sufficient, as investments are structured:

• Projects are often organized into decision groups
• no more than one option may be selected from each group
• at the same time, synergies or dependencies arise between groups

The multiple-choice backpack task extends the model to include precisely this reality.

2. What is the multiple-choice rucksack task?

Formally, the multiple-choice backpack task describes an optimization problem in which:

• the total budget is limited,
• Investment options are available in disjoint groups,
• a maximum of one option may be selected from each group,
• the total value is to be maximized.

Applied to companies, this means

• Group = decision category (e.g. technology path, location, provider)
• Options = concrete investment alternatives
• Budget = capital, time, resources

3. Alternative assets: either-or decisions

Alternative assets are mutually exclusive. Typical examples:

• Purchase or leasing
• In-house development or purchase
• Technology A or technology B
• Location X or location Y

These decisions are often viewed in isolation. In reality, however, they compete for the same budget and influence other investment Investment decisions.

The multiple-choice rucksack task enforces structural discipline here: Only one option may be chosen from each alternative group - regardless of how attractive several options appear individually.

4. Complementary assets: Value is created through interaction

Complementary assets are even more complex. Their value is not additive, but conditional:

• software only develops value with the right hardware
• a production site only becomes efficient through logistics
• an acquisition is only effective with integration and IT

In classic business cases, such dependencies are often described qualitatively - but not calculated. This results in systematic misjudgements of benefit and risk.

5. Why linear evaluation fails here

Many investment decisions are based on:

• Individual ROI
• isolated business cases
• linear scorecards

These methods implicitly assume that the value of individual assets is independent of each other. This is precisely not the case with complementary and alternative assets.

The result is portfolios that look attractive on paper, but in reality are

• incomplete
• over-complex
• underfunded
• or strategically inconsistent

6. The multiple-choice rucksack task as a portfolio model

The MCKP model forces a decisive perspective:

It is not the quality of individual investments that determines success, but the combination of permissible options under budget constraints.

It systematically answers

• which alternatives must be suppressed
• which complementary assets should be selected together
• where budget has the highest overall impact

7. Why experience and Excel are not enough

The number of possible portfolios explodes with just a few decision groups:

• 8 groups with 4 options each →⁴⁸ = 65,536 combinations
• 10 groups with 5 options each →⁵¹⁰ ≈ 9.8 million combinations
• plus budget restrictions and dependencies → exponential explosion

Excel can add options, but cannot identify globally optimal combinations. Experience helps with the assessment of individual options - not with the evaluation of the entire solution space.

8. Strategic mistakes without combinatorial optimization

Without formal optimization, the following regularly occur

• inconsistent technology portfolios
• strategic dead ends
• Overinvestment in individual areas
• lack of resources for complementary building blocks

These errors are rarely due to individual incorrect assumptions, but rather to methodological limitations of the decision-making logic.

9. Governance and transparency effect

An often underestimated advantage of multiple-choice backpack logic is its transparency:

• Decision rules are explicit
• Alternatives are comprehensibly excluded
• Budget allocation can be justified
• Decisions can be audited and governed

For the Executive Board, Supervisory Board and investors in particular, this makes it clear why certain options were chosen and others were deliberately not chosen.

Conclusion

The multiple-choice rucksack task precisely describes the reality of modern investment decisions: limited budgets, alternative options and complementary assets.

Companies that continue to make linear decisions optimize individual measures - and lose overall impact. Companies that understand investment as a combinatorial optimization problem do not maximize value by saving, but through structured selection and consistent portfolios.

The decisive question is therefore not: Which investment is the best?
But rather: Which combination of permissible investments generates the highest overall benefit under real restrictions?

Have your investment portfolio structure optimized now and create a higher overall benefit!